Oh, absolutely they do. That's why currency is called "banknotes". The banks printed them. Each bank printed their own banknotes. You can find their images in numismatics books.
In 1914, the government took over the function of printing banknotes. But the banks still print money. We just call them "cashier's checks", "money orders" and "travelers' checks". But more normally, they simply credit your account with created money.
> If stuff is harder to make or is more rare, prices will go up irrespective of money supply.
That isn't inflation, because extra dollars are not being created. For example, tomatoes going up in price in the store because of crop failure isn't inflation, because more money doesn't appear in your pocket to pay the premium. What happens is you wind up with fewer dollars in your pocket, meaning you buy less of other things, and with the annoying Law of Supply and Demand, the prices of those other things drop.